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Mirrors > Home > ILE Home > Th. List > 3sstr3i | GIF version |
Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
3sstr3.1 | ⊢ 𝐴 ⊆ 𝐵 |
3sstr3.2 | ⊢ 𝐴 = 𝐶 |
3sstr3.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3sstr3i | ⊢ 𝐶 ⊆ 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr3.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | 3sstr3.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
3 | 3sstr3.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | sseq12i 3130 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷) |
5 | 1, 4 | mpbi 144 | 1 ⊢ 𝐶 ⊆ 𝐷 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ⊆ wss 3076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-in 3082 df-ss 3089 |
This theorem is referenced by: (None) |
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